Nonarchimedean superrigidity of solvable S-arithmetic groups

2014/01/21 by Dave Witte Morris, Morris, Dave Witte, Daniel Studenmund +1
Mathematics · #20G30 (Primary) 22E40 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20G30 #msc:22E40

paper · pdf · doi:10.48550/arxiv.1401.5142

16 pages; corrected errors in the statements of the results over ground fields that are a proper extension of Q, and revised the abstract and introduction to state these general results instead of only over Q

arxiv created 2014/06/17 · arxiv updated 2014/06/18

Abstract

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any local field and alpha is any homomorphism from Gamma to GL(n,L), then alpha virtually extends (modulo a bounded error) to a continuous homomorphism defined on some finite-index subgroup of H. In the special case where F is the field of rational numbers, the real-rank of G is 0, and Gamma is Zariski-dense in G, we may let H = G_S. We also point out a generalization that does not require G to be solvable.

Related

• Archimedean superrigidity of solvable S-arithmetic groups
• Automorphism groups of solvable groups of finite abelian ranks
• Solvability and nilpotency for algebraic supergroups
• Tamely ramified subfields of division algebras
• Superrigid subgroups of solvable Lie groups